The Equation f′′′ + ff′′ + g(f′) = 0 and the Associated Boundary Value Problems

Abstract

We study the concave and convex solutions of the third order similarity differential equation f + ff + g(f ) = 0, and especially the ones that satisfies the boundary conditions f (0) = a, f (0) = b and f (t) → λ as t → +∞, where λ is a root of the function g. According to the sign of g between b and λ, we obtain results about existence, uniqueness and boundedness of solutions to this boundary value problem, that we denote by (P g;a,b,λ ). In this way, we pursue and complete the study done in 2008.Mathematics Subject Classification (2010). 34B15, 34C11, 76D10.

Similarity solutions of mixed convection boundary-layer flows in a porous medium

“…Without further assumptions on g, it is hopeless to have more precise results. Nevertheless, the results of [9] generalize the ones of [12] and some of [19] about mixed convection problems.…”

“…In the case where 0 < β 1, we are able to get complete results (and so we improve the results of [19]), while we only have partial results for β > 1. On several occasions, we will use the results of [9], that sometimes we re-demonstrate, in our particular case, for the convenience of the reader.…”

“…The boundary value problems associated to the general equation (2), with the condition that f ′ tends to λ at infinity have been studied in [13] and in [9]. Let us notice that, if g(λ) = 0, then these boundary value problems do not have any solutions, and thus we must assume that g(λ) = 0 to have solutions.…”

“…This condition at infinity can be, either f ′ (t) → λ as t → +∞, or f ′ (t) ∼ A t ν as t → +∞, where A and ν are some positive constants, or also |f | is of polynomial growth at infinity. For more details, we refer to the introduction of [9] and to the references therein.Proof -This follows immediatly from the uniqueness of solutions of initial value problem. Cf.[9], Proposition 3.1, item 3.…”

On the solutions of a boundary value problem arising in free convection with prescribed heat flux

“…Indeed, if x ∈ (0, 1) is such that w(x) > 0, w (x) = 0, then using (4.1) we have has at most one bounded concave solution, if 0 ≤ g(x) ≤ x 2 , see [4], and this result is one of the reasons for which we hope that this conjecture holds. On the other hand, let us notice that we recover the results of Tsai and Wang [15], in a totally different way, and perhaps more directly.…”